Abstract

Chiral separation is a crucial step in many chemical synthesis processes, particularly for pharmaceuticals. Here we present a novel method for the realization of both separating and trapping of enantiomers using the dielectric photonic crystal (PhC) slabs, which possess quasi-fourfold degenerate Bloch modes (overlapping double degenerate transverse-electric-like and transverse-magnetic-like modes). Based on the designed structure, a large gradient of optical chirality appears near the PhC slab, leading to the extreme enhancement of chiral optical forces about 3 orders of magnitude larger than those obtained with circularly polarized lights. In this case, our method provides a reference for realizing all-optical enantiopure syntheses.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Chirality, which refers to structures lacking any mirror symmetry planes, is a very intriguing property of enantiomers. For example, many biologically active molecules are chiral, which plays a pivotal role in biochemistry and the evolution of life itself. When a protein loses its original chirality, it can become toxic to cells [1]. Thus, separating and trapping of enantiomers have gained traction in the biochemical and pharmaceutical industries [2,3]. Since enantiomers are identical in their chemical composition and scalar physical properties, prevailing chemical methods remain inefficient in separation and purification of enantiomer mixtures of chiral biomolecules [4].

Optical manipulation is usually used to separate enantiomers [57] where the optical force acted on the particle with different chirality is different due to the opposite sign of their chiral polarizabilities [813]. In general, the light force is very weak; such a method is only suitable for the separation of special enantiomers. Plasmonic optical tweezers increasing the optical forces by enhancing the near-field gradients have emerged as a powerful strategy to separate enantiomers [1425]. Selective optical capture of sub-10 nm chiral particles has been achieved by using an achiral plasmonic tweezer [14]. However, because of the small size of the plasmonic tweezer and the lack of strong chiral light-matter interaction, the amount of the separated chiral particles should be limited. In addition, plasmonic optical tweezers can also induce a giant photo-thermal effect, which leads to denaturation of the chiral particles [26,27].

On the other hand, optical chirality [$C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}{\mathop{\rm Im}\nolimits} ({\textbf{E}^ \ast } \cdot \textbf{H})$] has been developed to enhance the interaction between chiral lights and matters [2841]. Recent investigations have shown that superchiral surface waves exhibit many advantages for all-optical enantiomer separation [42]. Based on such an approach, chiral optical forces can be improved by 2 orders of magnitude than those obtained with circularly polarized lights (CPLs) [42]. To further increase the chiral light-matter interaction, our recent investigations have proved that optical fields with a much larger optical chirality can be achieved using photonic crystal (PhC) slabs that utilized the coupling and hybridizing of the transverse-electric (TE)-like and transverse-magnetic (TM)-like modes [4347]. A large portion of electromagnetic (EM) energy concentrated in the hole can be used to perform surface enhanced CD and Raman optical activity of chiral molecules [43,44,48].

Motivated by the above investigations, in this work we propose a scheme to separate and trap enantiomers with dielectric PhC slabs. Our designed PhC slabs possess the quasi-fourfold degenerate mode. In such a case, a large gradient of optical chirality can be generated near the PhC slabs. Thus, under the excitation of CPLs, the chiral optical force can push one of the enantiomers toward certain regions of the PhC slab, but let the other one move away. Our theory indicates that achiral PhC slabs, providing a possible route toward all-optical enantiopure syntheses, can mediate the chiral enantiomers.

2. Theory and method

We consider a dielectric PhC slab with a square array of cylindrical holes, as shown in Fig. 1(a). The PhC slab is placed on the hollow SiO2 substrate with a refractive index of 1.46. Figure 1(b) shows a unit cell of the PhC slab. The lattice constant, the radius of the nanohole and the thickness of the PhC slab are marked by a, r and t, respectively. To ensure the symmetry of the system, the PhC slab with a refraction index n = 2.02 (Si3N4) is immersed in a medium with the refraction index being nb = 1.46. To fulfill the chiral separation, the selected medium is in terms of not only the refractive index but also density, viscosity, biocompatibility. For example, the CCl4 (Carbon tetrachloride) solution is a good choice. Considering the periodicity of the PhC slab in the XY plane, the resonant properties of the PhC slab can be described by a photonic band structure, as shown in Figs. 1(c) and 1(d). Figures 1(c) and 1(d) show the dispersion relation of the eigenfrequency as a function of Bloch wave vector normalized by 2π/a at t = 280 nm and t = 326.2 nm, respectively. Here, other structural parameters are chosen to be a = 336 nm and r = 80 nm. Because the PhC slab is symmetric with respect to the plane z = 0 nm, the Bloch modes can be classified into TE-like and TM-like modes, which are marked in Figs. 1(c) and 1(d) with blue squares and orange triangles, respectively. At the Brillouin zone center (Γ point), the structures support both double degenerate TE-like and TM-like modes. We enlarge the plot near the double degenerate TE-like and TM-like modes, as illustrated in Figs. 1(c) and 1(d). The blue squares and orange triangles in the insets correspond to the double degenerate TE-like and TM-like modes, respectively. We find that the double degenerate TE-like and TM-like modes are separated at t = 280 nm, and the two different kinds of modes become coincident, giving rise to a quasi-fourfold degenerate mode at t = 326.2 nm.

 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of the PhC slab placed on the SiO2 substrate. There is a gap between the PhC slab and the substrate. (b) Geometry and coordinate of a unit cell for the PhC slab. The lattice constant, the radius of the nanohole and the thickness of the PhC slab are marked by a, r and t, respectively. (c) Dispersion relation in ΓΧ and ΓΜ directions as a function of |k|a/2π. Where k is the Bloch wave vector. The structural parameters are chosen as a = 395 nm, r = 60 nm, t = 280 nm. Moreover, c is the vacuum speed of light. (d) Dispersion relation in ΓΧ and ΓΜ directions as a function of Bloch wave vector normalized by 2π/a. The structural parameters are chosen as a = 395 nm, r = 60 nm, t = 326.2 nm. (e)-(f) The distributions of electric ($\widetilde {\textbf E}$) and magnetic ($\widetilde {\textbf H}$) fields for the double degenerate TM-like and TE-like modes in the YZ plane of the unit cell, respectively. The blue arrows denote the real part of the electric and magnetic fields. In addition, the black solid line frames the Si3N4 of the unit cell. The parameters of the PhC slab are chosen as a = 395 nm, r = 60 nm, and t = 280 nm. (g)-(i) The averaged enhancements of electric, magnetic and optical chirality in the cylindrical nanohole of the PhC slab with the cases of t = 280 nm and 326.2 nm.

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In order to explain the quasi-fourfold degenerate mode more clearly and show its advantages, in Figs. 1(e) and 1(f), we plot the distributions of electric and magnetic fields for the TM-like and TE-like modes in a unit cell, respectively. The blue arrows in Figs. 1(e) and 1(f) denote the real part of the electric and magnetic fields. We note that the electric (magnetic) fields for the TE-like (TM-like) mode are mainly concentrated at the center region of the nanohole while the electric (magnetic) fields for the TM-like (TE-like) mode circulate around the dielectric region. According to the definition of optical chirality, $C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}{\mathop{\rm Im}\nolimits} ({\textbf{E}^ \ast } \cdot \textbf{H})$, it should reach maximum when the electric field and magnetic field are both significantly enhanced at the same region, and have parallel components with the phase difference being π/2. From the distributions of $\widetilde {\textbf E}$ and $\widetilde {\textbf H}$ fields in Figs. 1(e) and 1(f), it is expected that the PhC slab would not give rise to extremely large optical chirality if only the double degenerate TM-like or TE-like mode is excited alone. This is because, firstly, the region where the electric fields have large values does not overlap with that of the magnetic fields, and, more importantly, the electric fields are nearly perpendicular with the magnetic fields. These issues can be solved by the degenerate state of TM-like and TE-like modes (the quasi-fourfold degenerate mode), which can be simultaneously excited, making the optical chirality of near-fields be extremely enhanced.

Figures 1(g)–1(i) show the averaged enhancements of electric field intensity ($\int_V {{{|\textbf{E} |}^2}} /{|{{\textbf{E}_0}} |^2}dV$), magnetic field intensity ($\int_V {{{|\textbf{H} |}^2}} /{|{{\textbf{H}_0}} |^2}dV$) and optical chirality ($\int_V {C/{C_0}} dV$) in the hole of the PhC slab under the CPL excitation along the z-axis. Here, V denotes the volume of the nanoholes of the PhC slab; E and H are the electric and magnetic fields with the existence of PhC slabs. E0, H0 and C0 represent the electric field, magnetic field and optical chirality of the CPL in the background, respectively. The red (circle) and blue (square) lines correspond to the cases with t = 280 nm and 326.2 nm. It is worthy to note that the two peaks of the case with t = 280 nm (red circle lines) are attributed to the excitation of the double degenerate TM-like modes (at λ = 705.2 nm) and TE-like modes (at λ = 710.3 nm). To further understand the PhC slab with different Bloch modes, we show the distributions of electric and magnetic fields in the unit cell of the PhC slab under the excitation of CPL in Supplement 1. Comparing the enhancement factors of optical chirality with t = 280 nm and 326.2 nm in Fig. 1(i), it is clearly shown that the enhancement factor of the system sustaining quasi-fourfold degenerate mode is much larger than that with separated double degenerate TE-like and TM-like modes. All numerical results are calculated by COMSOL. The eigenfrequency simulation is used for the dispersion relation calculating, and the wavelength domain simulation is used for the optical fields and the gradient of optical chirality calculating.

Now, we put a chiral particle on the PhC slab. We focus on the chiral nanoparticles with the radius of 10 nm. Actually, there are numerous synthetic chiral nanoparticles that are around 10 nm, such as carbon nanotubes, chiral nanocrystals, chiral quantum dots, and DNA-assembled nanoparticles [4952]. This approach assumes that the chiral specimens do not perturb the fields, which is typical for specimens that are much smaller than the aperture and within the Rayleigh regime [14]. Here, the size of the chiral particle considered in our work is much smaller than the incident wavelength (λ = 719.8 nm), where the polarizability of the chiral particle of 10 nm is considered for all numerical results. The chiral particle can be modeled as a pair of interacting electric and magnetic dipoles. When the EM waves act on the chiral particle, the induced dipole moments of such a chiral particle can be expressed as [13,53]:

$$\left[ {\begin{array}{c} \textbf{p}\\ \textbf{m} \end{array}} \right] = \left[ {\begin{array}{cc} {{\alpha_{\textrm{ee}}}}&{i{\alpha_{\textrm{em}}}}\\ { - i{\alpha_{\textrm{em}}}}&{{\alpha_{\textrm{mm}}}} \end{array}} \right]\left[ {\begin{array}{c} \textbf{E}\\ \textbf{H} \end{array}} \right],$$
where p and m represent the electric and magnetic dipole moments, αee, αmm, and αem are the electric, magnetic and electromagnetic polarizabilities of the chiral particle, respectively. The electromagnetic polarizability αem is closely related to the chirality parameter κ of the particle as [14,54]:
$${\alpha _{\textrm{em}}} ={-} 12\pi r_\textrm{p}^3\frac{{\kappa \sqrt {{\mu _0}{\varepsilon _0}} }}{{({\varepsilon _\textrm{r}} + 2{\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}},$$
where ${\varepsilon _0}$ and ${\mu _0}$ are the permittivity and permeability of vacuum, rp is the radius of the chiral particle, ${\varepsilon _\textrm{r}}$ and ${\mu _\textrm{r}}$ are the relative permittivity and permeability of the chiral particle, ${\varepsilon _{\textrm{rm}}}$ is the relative permittivity of the medium where the particle is embedded. The parameter κ is dimensionless, and is used to estimate the degree of chirality of chiral materials. It can be seen from Eq. (2) that the sign of αem depends on κ. In this case, ${\alpha _{\textrm{em}}} = 0$ corresponds to an achiral particle. In addition, electric and magnetic polarizabilities can be expressed as [14,54]:
$${\alpha _{\textrm{ee}}} = 4{\varepsilon _0}\pi r_\textrm{p}^3\frac{{({\varepsilon _\textrm{r}} - {\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}}{{({\varepsilon _\textrm{r}} + 2{\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}}, $$
$${\alpha _{\textrm{mm}}} ={-} 4{\mu _0}\pi r_\textrm{p}^3\frac{{{\kappa ^2}}}{{({\varepsilon _\textrm{r}} + 2{\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}}. $$

It is clearly shown that the sign of electric polarizability αee or magnetic polarizability αmm is independent of κ. In general, the optical force acting on a dipolar chiral particle can be written as [5557]:

$$\left\langle \textbf{F} \right\rangle = \frac{1}{2}{\textrm{Re}} \left[ {(\nabla {\textbf{E}^ \ast }) \cdot \textbf{p + (}\nabla {\textbf{H}^ \ast }) \cdot \textbf{m - }\frac{{c{k^4}}}{{6\pi \sqrt {{\varepsilon_{\textrm{rm}}}} }}(\textbf{p} \times {\textbf{m}^ \ast })} \right] .$$

Substitute Eq. (1) into Eq. (5), we can obtain the expression of optical force as [13]:

$$\begin{array}{l} \left\langle \textbf{F} \right\rangle = \nabla U + \sigma \frac{{\left\langle \textbf{S} \right\rangle }}{c} - {\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{em}}}]\nabla \times \left\langle \textbf{S} \right\rangle + \frac{c}{{{\varepsilon _{\textrm{rm}}}}}{\sigma _\textrm{e}}\nabla \times \left\langle {{\textbf{L}_\textrm{e}}} \right\rangle + \frac{c}{{{\varepsilon _{\textrm{rm}}}}}{\sigma _\textrm{m}}\nabla \times \left\langle {{\textbf{L}_\textrm{m}}} \right\rangle \\ + \omega {\gamma _\textrm{e}}\left\langle {{\textbf{L}_\textrm{e}}} \right\rangle + \omega {\gamma _\textrm{m}}\left\langle {{\textbf{L}_\textrm{m}}} \right\rangle \textrm{ + }\frac{{c{k^4}}}{{12\pi \sqrt {{\varepsilon _{\textrm{rm}}}} }}{\kern 1pt} {\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{ee}}}\alpha _{\textrm{mm}}^ \ast ]{\mathop{\rm Im}\nolimits} [\textbf{E} \times {\textbf{H}^ \ast }] \end{array},$$
where $U = \frac{1}{4}({\textrm{Re}} [{\alpha _{\textrm{ee}}}]{|\textbf{E} |^2} + {\textrm{Re}} [{\alpha _{\textrm{mm}}}]{|\textbf{H} |^2} - 2{\textrm{Re}} [{\alpha _{\textrm{em}}}]{\mathop{\rm Im}\nolimits} [\textbf{H} \cdot {\textbf{E}^ \ast }])$ depends on the gradient of the field, $\left\langle \textbf{S} \right\rangle = \frac{1}{2}{\textrm{Re}} [\textbf{E} \times {\textbf{H}^ \ast }]$ is the time-averaged Poynting vector, $\left\langle {{\textbf{L}_\textrm{e}}} \right\rangle = \frac{{{\varepsilon _0}{\varepsilon _{\textrm{rm}}}}}{{4\omega i}}\textbf{E} \times {\textbf{E}^ \ast }$ and $\left\langle {{\textbf{L}_\textrm{m}}} \right\rangle = \frac{{{\mu _0}}}{{4\omega i}}\textbf{H} \times {\textbf{H}^ \ast }$ are the time-averaged spin densities, ${\sigma _\textrm{e}} = \frac{k}{{{\varepsilon _0}\sqrt {{\varepsilon _{\textrm{rm}}}} }}{\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{ee}}}]$, ${\sigma _\textrm{m}} = \frac{{k\sqrt {{\varepsilon _{\textrm{rm}}}} }}{{{\mu _0}}}{\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{mm}}}]$, $\sigma \textrm{ = }{\sigma _\textrm{e}} + {\sigma _\textrm{m}} - \frac{{{c^2}{k^4}}}{{6\pi \sqrt {{\varepsilon _{\textrm{rm}}}} }}({\textrm{Re}} [{\alpha _{\textrm{ee}}}\alpha _{\textrm{mm}}^ \ast ] + {\alpha _{\textrm{em}}}\alpha _{\textrm{em}}^ \ast )$, ${\gamma _\textrm{e}} ={-} 2\omega {\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{em}}}] + \frac{{c{k^4}}}{{3\pi {\varepsilon _0}{\varepsilon _{\textrm{rm}}}\sqrt {{\varepsilon _{\textrm{rm}}}} }}{\textrm{Re}} [{\alpha _{\textrm{ee}}}\alpha _{\textrm{em}}^ \ast ]$ and ${\gamma _\textrm{m}} ={-} 2\omega {\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{em}}}] + \frac{{c{k^4}}}{{3\pi {\mu _0}\sqrt {{\varepsilon _{\textrm{rm}}}} }}{\textrm{Re}} [{\alpha _{\textrm{mm}}}\alpha _{\textrm{em}}^ \ast ]$ are the coefficients with the dimension of a cross-section.

The first term in Eq. (6) corresponds to the gradient force, which plays a major role in the design process. We denote the gradient force as Fg, Fg can be expressed as:

$${\textbf{F}_\textrm{g}} = \frac{1}{4}({\textrm{Re}} [{\alpha _{\textrm{ee}}}]\nabla {|\textbf{E} |^2} + {\textrm{Re}} [{\alpha _{\textrm{mm}}}]\nabla {|\textbf{H} |^2} - 2{\textrm{Re}} [{\alpha _{\textrm{em}}}]\nabla {\mathop{\rm Im}\nolimits} [\textbf{H} \cdot {\textbf{E}^ \ast }]),$$
which can be divided into dielectric gradient force Fd and chiral gradient force Fk, the expressions can be written as:
$${\textbf{F}_\textrm{d}} = \frac{1}{4}{\textrm{Re}} [{\alpha _{\textrm{ee}}}]\nabla {|\textbf{E} |^2} + \frac{1}{4}{\textrm{Re}} [{\alpha _{\textrm{mm}}}]\nabla {|\textbf{H} |^2} , $$
$${\textbf{F}_\textrm{k}} ={-} \frac{{{\textrm{Re}} [{\alpha _{\textrm{em}}}]}}{2}\nabla {\mathop{\rm Im}\nolimits} [\textbf{H} \cdot {\textbf{E}^ \ast }]. $$

According to the optical chirality ($C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}{\mathop{\rm Im}\nolimits} ({\textbf{E}^ \ast } \cdot \textbf{H})$), the gradient of optical chirality $\nabla C$ can be written as:

$$\nabla C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}\nabla {\mathop{\rm Im}\nolimits} (\textbf{E}_x^ \ast{\cdot} {\textbf{H}_x} + \textbf{E}_y^ \ast{\cdot} {\textbf{H}_y} + \textbf{E}_z^ \ast{\cdot} {\textbf{H}_z}). $$

The subscripts x, y, and z are the coordinate components. It can be seen from Eq. (9), the chiral gradient force is directly related to the gradient of optical chirality $\nabla C$. Because the signs of chiral gradient forces are different for enantiomers, hence, when the chiral gradient force is larger than the dielectric gradient, the separation of enantiomers can be realized.

3. Numerical results and discussion

As shown by Eq. (6), the optical force is not only related to the gradients of electric and magnetic fields, but also depends on the Poynting vector and spin angular momentum of the fields. Firstly, we focus on the contribution of each term of the optical force on the chiral particle. The red solid and black dash lines in Fig. 2(a) show the total optical force F in Eq. (6) and the gradient force Fg in Eq. (7) under the excitation by the left-handed circularly polarized (LCP) light, respectively. The target chiral compounds are modeled as spherical nanoparticles with the refractive index nr = 1.45. The chiral particle is placed at the position as shown in the inset of Fig. 2(a). The radius of the chiral particle is 10 nm and the particle chirality is taken as κ = +0.5, such a chirality value is achievable with synthetic chiral structures, including the chiral nanoparticles [8,9,14,28,42,52]. It is seen clearly that the magnitude of the gradient force Fg is nearly identical to the magnitude of the total optical force F. Such a phenomenon is similar to many other optical tweezers [15,5861]. Hence, the expression of optical force near the PhC slab can be simplified as [13,14]:

$$\left\langle \textbf{F} \right\rangle \approx {\textbf{F}_\textrm{g}} = {\textbf{F}_\textrm{d}} + {\textbf{F}_\textrm{k}}$$

Based on the discussion above, we know that the PhC slab with quasi-fourfold degenerate mode can produce strong optical chirality under the excitation of CPL. While the PhC slab with a single double degenerate TM-like or TE-like mode can only produce strong electric or magnetic fields. In this case, the large and uneven distribution of the optical chirality near the PhC slab with quasi-fourfold degenerate mode can make the gradient of optical chirality be extremely enhanced. To clearly show the difference between the structures with and without the degenerate modes, the PhC slab with different thicknesses, that is t = 326.2 nm (t = 280 nm) corresponds to the system with (without) the quasi-fourfold degenerate mode, are investigated. In Figs. 2(b) and 2(c), we calculate the magnitudes of Fd and Fk for cases with t = 326.2 nm and 280 nm, respectively. As shown by Eqs. (8) and (9), the dielectric gradient force Fd is related to the gradients of electric field and magnetic field, and the chiral gradient force Fk is related to the gradient of optical chirality. The orange solid and blue dash lines in Figs. 2(b) and 2(c) correspond to the magnitude of Fk and Fd. We find that the magnitude of Fk is around 4 times larger than Fd at the resonance wavelength of the quasi-fourfold degenerate mode (λ = 719.8 nm) due to the large gradient of optical chirality. In contrast, the magnitude of Fd is always larger than Fk at the resonance wavelength of separated TM-like (λ = 705.2 nm) or TE-like (λ = 710.3 nm) mode. Hence, the PhC slab sustaining the quasi-fourfold degenerate modes possesses a much better role to realize chiral separations.

 figure: Fig. 2.

Fig. 2. (a) Numerically calculated the total optical force F and the gradient force Fg acting on the chiral particles with the excitation of LCP light. The radius of the chiral particle is 10 nm. Moreover, the chiral particle is placed 20 nm above the PhC slab. (b) The magnitude of dielectric gradient force Fd and chiral gradient force Fk with the thickness of the PhC slab t = 326.2 nm. (c) The magnitude of dielectric gradient force Fd and chiral gradient force Fk with the thickness of the PhC slab t = 280 nm. The radius of the chiral particle is 10 nm with the refractive index nr = 1.45 and the particle chirality is κ = +0.5. The other structural parameters are chosen as a = 336 nm, r = 80 nm.

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To further illustrate the chiral separation mechanism, Fig. 3(a) presents the intensity distribution and direction (marked by the white arrows) of optical forces in the YZ plane for cases with κ = +0.5 and κ = −0.5, respectively. Here, the thickness of the PhC slab is t = 326.2 nm which has a quasi-fourfold degenerate mode. When the PhC slab is illuminated by the LCP light along the z-axis, the optical force can push the chiral particles with the chirality parameter κ = +0.5 toward the nanohole, but let the chiral particles with κ = −0.5 move away from the nanohole. Thus, the chiral particles with the chirality parameter κ = +0.5 are trapped inside the nanohole of the PhC slab. Moreover, the chirality of the trapped particles is opposite when the incident LCP light is changed to the right-handed circularly polarized (RCP) light. For comparison, in Fig. 3(b), we plot the distributions of optical forces in the unit cell for the case of t = 280 nm. Here, the optical force acted on particles with different chirality have the same directions, and cannot be used for chiral separations. Consequently, the PhC slab with the quasi-fourfold degenerate mode can separate a pair of enantiomers while capturing one of the enantiomers with specific chirality, which cannot be achieved by the PhC slab with separated TM-like and TE-like modes.

 figure: Fig. 3.

Fig. 3. (a) and (b) Calculated optical forces for both enantiomers (κ = ±0.5) in the unit cell with the excitation of LCP light. The gray areas represent Si3N4 material. Moreover, the black solid line frames the Si3N4 of the unit cell. The non-dielectric regions include the nanohole and the range of 360 nm above and below the unit cell. The color map shows the magnitude of the optical force, and the white arrows show the directions of the optical force. The optical force along the z-axis (-z-axis) is marked by the upward (downward) arrows. The excitation wavelengths with the thickness of the PhC slab (a) t = 326.2 nm and (b) t = 280 nm are 719.8 nm and 705.2 nm, respectively. (c) The trapping potential along the z-axis for both enantiomers when the thickness of the PhC slab is 326.2 nm under the excitation of LCP light. Here, the excitation wavelength is 719.8 nm. (d) The trapping potential along the z-axis for both enantiomers when the thickness of the PhC slab is 280 nm under the excitation of LCP light. Here, the excitation wavelength is 705.2 nm. The red solid and blue dashed lines represent the trapping potential of the chiral particle with the chirality being 0.5 and −0.5, respectively. The trapping potential is calculated when the power of incident light is 100 mW in the unit cell.

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Indeed, it is important to ensure that the particles are stably trapped. To trap a particle stably, the trapping potential must be larger than 10 kBT, which is enough to oppose the Brownian motion of the nanoparticles suspended in solution [14,58]. We have calculated the one-dimensional trapping potential of the chiral particle along the z-axis. Figure 3(c) shows the trapping potential for both enantiomers when the thickness of the PhC slab is 326.2 nm under the excitation of LCP light. Here, the trapping potential is calculated when the power of incident light is 100 mW in the unit cell. The red solid and blue dashed lines represent the trapping potential of the chiral particle with the chirality being 0.5 and −0.5, respectively. As we can see, the PhC slab with quasi-fourfold degenerate mode provides a positive trapping potential for the chiral particle with the chirality being −0.5, while for the chiral particle with the chirality being 0.5, the trapping potential is negative at the same location. This distinct trapping potential stably captures only the chiral particle with the chirality being 0.5 at the range of 185 nm above the PhC slab, keeping the chiral particles with the chirality being −0.5 away from the nanohole. To trap the other handedness, one may simply reverse the incidence polarization to RCP light. According to the previous work [14,25], the chiral particles are generally placed at the position of 20 nm above the structure. Here, the chiral particles placed at the position of 20 nm above the PhC slab can be trapped by a trapping potential of more than 10 kBT using 5 mW (33 mW/µm2) of the illumination power. Figure 3(d) shows the trapping potential for both enantiomers when the thickness of the PhC slab is 280 nm. Without quasi-fourfold degenerate mode, the PhC slab provides both positive trapping potentials for chiral particles with different chirality. This is the reason that the chiral enantiomers cannot be separated by using the separated TM-like and TE-like modes.

To clarify the origin of the above phenomena, in Figs. 4(a) and 4(b), we plot the distributions of the near-field gradient of optical chirality with the thicknesses of PhC slabs being t = 326.2 nm and 280 nm, respectively. Here, the red arrows denote the directions of $\nabla C$. This shows clearly that the chiral gradients are in opposite directions on two sides of the PhC slab, and both directions are towards the nanohole. We note that the gradient of optical chirality $\nabla C$ for the system sustaining quasi-fourfold degenerate mode is much larger than that with separated double degenerate TE-like and TM-like modes. Based on such a giant gradient of optical chirality, the particles with positive (negative) κ can be trapped under the excitation of LCP (RCP) light.

 figure: Fig. 4.

Fig. 4. (a) Near-field distribution of the gradient of optical chirality with the thickness of the PhC slab t = 326.2 nm at the excitation wavelength λ = 719.8 nm. (b) Near-field distribution of the gradient of optical chirality with the thickness of the PhC slab t = 280 nm at the excitation wavelength λ = 705.2 nm. The color map shows the magnitude of the gradient of optical chirality $\nabla C$ and the red arrows denote the directions of $\nabla C$.

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In the above analysis, we only chose a fixed value for the chirality of manipulated particles. In the following, we discuss the influence of the magnitude for the particle chirality on the separation of enantiomers. In Figs. 5(a) and 5(b), we plot the total optical force F (red solid lines), the dielectric gradient component Fd (orange long dash lines) and the chiral gradient component Fk (blue dash lines) of the PhC slab with t = 326.2 nm and 280 nm as functions of chirality κ under the excitation of LCP light, respectively. Here, the sign of the force represents its direction. The positive (negative) sign means that the direction of the optical force is along the z-axis (–z-axis). We find that the magnitude of Fd is larger than Fk when κ ranges from 0 to ±0.01 for the PhC slab possessing quasi-fourfold degenerate modes, as shown in Fig. 5(a). This means that the chiral particles cannot be trapped by such a structure within this range. Additionally, as for the PhC slab with separated double degenerate TE-like and TM-like modes, the magnitude of ${{\textbf F}_\textrm{d}}$ is always larger than ${{\textbf F}_\textrm{k}}$ with κ =[−0.2, 0.2], as shown in Fig. 5(b). In such a case, the PhC slab with separated double degenerate TE-like and TM-like modes cannot trap the chiral particles. It is worthy to note that, although the PhC slab with the quasi-fourfold degenerate mode can trap the chiral particles with the chirality greater than 0.01 (LCP light excitations) of less than −0.01 (RCP light excitations), such value of chirality is still much larger than that of natural chiral molecules whose chirality is extremely small (about 10−5) [54]. Therefore, in order to achieve the capture of the natural chiral material, we need to design the structure that can further increase the gradient of the optical chirality while reducing the gradient of the energy density for both electric and magnetic fields.

 figure: Fig. 5.

Fig. 5. (a) Optical force F and its components Fd and Fk acting on the chiral particle as a function of chirality κ under the excitation of LCP light at λ = 719.8 nm. The thickness of the PhC slab is 326.2 nm. (b) Optical force F and its components Fd and Fk acting on the chiral particle as a function of chirality κ under the excitation of LCP light at λ = 705.2 nm. The thickness of the PhC slab is 280 nm. The red solid, orange long dash and blue dash lines correspond to the forces F, Fd and Fk respectively. Other parameters in Fig. 5 are the same as Fig. 2. Here, the sign of the force represents its direction. The positive (negative) sign means that the direction of the force is along the + z-axis (–z-axis).

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4. Summary

We have demonstrated that a large gradient of optical chirality can be generated due to the large and uneven distribution of the optical chirality on the PhC slab with a quasi-fourfold degenerate mode. Because of the large gradient of optical chirality, the chiral gradient force can be several times larger than the dielectric gradient force and plays a key role in chiral separating and trapping. As indicated above, the optical forces push one kind of enantiomers toward the PhC slab and trap it, but let the other away from the sample. To trap chiral particles with the other handedness, one can simply change the handedness of the incidence CPLs. Although our study may provide a strategy for realizing all-optical enantiopure syntheses, the current method can only separate chiral particles with κ being around 10−2, and the method for separating natural chiral molecules, whose chirality is extremely small (about 10−5), should be further investigated. Actually, it is worthy to note that the sensitivity of chiral separation depends on the ratio of the chiral gradient force Fk and the dielectric gradient force Fd. In this case, we proposed two promising pathways for chiral molecules separations. Refractive index matching can improve the enantioselectivity because it reduces the contribution of the dielectric gradient force Fd while maintaining the differences between the chiral gradient force Fk. Hence, we can engineer the structure to make the refractive index of background media match with that of detected chiral molecules. On the other hand, we can design the PhC slab to sustain the much stronger resonance to further enhance the sensitivity of separation for chiral molecules. For example, the bound states in the continuum, which possesses an infinite Q-factor and can generate the extremely enhanced gradient of chiral near-fields, should improve the sensitivity for chiral separations.

Funding

National Key Research and Development Program of China (2017YFA0303800); National Natural Science Foundation of China (91850205).

Disclosures

The authors declare no conflicts of interest.

Supplemental document

See Supplement 1 for supporting content.

References

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6. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]  

7. M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4(1), 1768 (2013). [CrossRef]  

8. A. Hayat, J. P. Balthasar Müller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015). [CrossRef]  

9. A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013). [CrossRef]  

10. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16(1), 013020 (2014). [CrossRef]  

11. G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014). [CrossRef]  

12. D. S. Bradshaw and D. L. Andrews, “Electromagnetic trapping of chiral molecules: orientational effects of the irradiating beam,” J. Opt. Soc. Am. B 32(5), B25–B31 (2015). [CrossRef]  

13. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014). [CrossRef]  

14. Y. Zhao, A. A. E. Saleh, and J. A. Dionne, “Enantioselective optical trapping of chiral nanoparticles with plasmonic tweezers,” ACS Photonics 3(3), 304–309 (2016). [CrossRef]  

15. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). [CrossRef]  

16. N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12(2), 977–983 (2012). [CrossRef]  

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24. A. A. E. Saleh, S. Sheikhoelislami, S. Gastelum, and J. A. Dionne, “Grating-flanked plasmonic coaxial apertures for efficient fiber optical tweezers,” Opt. Express 24(18), 20593–20603 (2016). [CrossRef]  

25. Z. H. Lin, J. Zhang, and J. S. Huang, “Plasmonic elliptical nanoholes for chiroptical analysis and enantioselective optical trapping”, arXiv 2101.04453 (2021).

26. W. Zhang, T. Wu, R. Wang, and X. Zhang, “Surface-enhanced circular dichroism of oriented chiral molecules by plasmonic nanostructures,” J. Phys. Chem. C 121(1), 666–675 (2017). [CrossRef]  

27. A. Pham, A. Zhao, C. Genet, and A. Drezet, “Optical chirality density and flux measured in the local density of states of spiral plasmonic structures,” Phys. Rev. A 98(1), 013837 (2018). [CrossRef]  

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29. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010). [CrossRef]  

30. Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011). [CrossRef]  

31. H. K. Bisoyi and Q. Li, “Light-directed dynamic chirality inversion in functional self-organized helical superstructures,” Angew. Chem., Int. Ed. 55(9), 2994–3010 (2016). [CrossRef]  

32. Y. Liu, W. Zhao, Y. Ji, R. Y. Wang, X. Wu, and X. D. Zhang, “Strong superchiral field in hot spots and its interaction with chiral molecules,” Europhys. Lett. 110(1), 17008 (2015). [CrossRef]  

33. T. Wu, X. H. Zhang, R. Y. Wang, and X. D. Zhang, “Strongly enhanced Raman optical activity in molecules by magnetic response of nanoparticles,” J. Phys. Chem. C 120(27), 14795–14804 (2016). [CrossRef]  

34. C. S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. A. Dionne, “Enhancing enantioselective absorption using dielectric nanospheres,” ACS Photonics 4(2), 197–203 (2017). [CrossRef]  

35. G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017). [CrossRef]  

36. W. Zhang, T. Wu, R. Wang, and X. Zhang, “Amplification of molecular chiroptical effect by low-loss dielectric nanoantennas,” Nanoscale 9(17), 5701–5707 (2017). [CrossRef]  

37. T. Wu, W. Zhang, R. Wang, and X. Zhang, “A giant chiroptical effect caused by the electric quadrupole,” Nanoscale 9(16), 5110–5118 (2017). [CrossRef]  

38. M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019). [CrossRef]  

39. E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018). [CrossRef]  

40. F. Graf, J. Feis, X. Garcia-Santiago, M. Wegener, C. Rockstuhl, and I. Fernandez-Corbaton, “Achiral, helicity preserving, and resonant structures for enhanced sensing of chiral molecules,” ACS Photonics 6(2), 482–491 (2019). [CrossRef]  

41. J. Hu, M. Lawrence, and J. A. Dionne, “High quality factor dielectric metasurfaces for ultraviolet circular dichroism spectroscopy,” ACS Photonics 7(1), 36–42 (2020). [CrossRef]  

42. P. Giovanni, F. Marco, C. Michele, D. Lamberto, A. I. Maria, M. M. Onofrio, and B. Paolo, “Superchiral surface waves for all-optical enantiomer separation,” J. Phys. Chem. C 123(46), 28336–28342 (2019). [CrossRef]  

43. T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020). [CrossRef]  

44. T. Wu, S. S. Hou, W. X. Zhang, and X. D. Zhang, “Strong superchiral fields and an ultrasensitive chiral sensor of biomolecules based on a dielectric photonic crystal slab with air holes,” Phys. Rev. A 102(5), 053519 (2020). [CrossRef]  

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46. I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111(6), 060401 (2013). [CrossRef]  

47. S. Yu, H. S. Park, X. Piao, B. Min, and N. Park, “Low-dimensional optical chirality in complex potentials,” Optica 3(9), 1025–1032 (2016). [CrossRef]  

48. S. S. Hou, T. Wu, W. X. Zhang, and X. D. Zhang, “Strongly enhanced Raman optical activity of chiral molecules by vector exceptional points,” J. Phys. Chem. C 124(45), 24970–24977 (2020). [CrossRef]  

49. X. Wei, T. Tanaka, Y. Yomogida, N. Sato, R. Saito, and H. Kataura, “Experimental determination of excitonic band structures of single-walled carbon nanotubes using circular dichroism spectra,” Nat. Commun. 7(1), 12899 (2016). [CrossRef]  

50. A. Ben-Moshe, S. G. Wolf, M. B. Sadan, L. Houben, Z. Fan, A. O. Govorov, and G. Markovich, “Enantioselective control of lattice and shape chirality in inorganic nanostructures using chiral biomolecules,” Nat. Commun. 5(1), 4302 (2014). [CrossRef]  

51. K. Varga, S. Tannir, B. E. Haynie, B. M. Leonard, V. Dzyuba, J. Kubelka, and M. Balaz, “CdSe quantum dots functionalized with chiral, thiol-free carboxylic acids: unraveling structural requirements for ligand-induced chirality,” ACS Nano 11(10), 9846–9853 (2017). [CrossRef]  

52. A. J. Mastroianni, S. A. Claridge, and A. P. Alivisatos, “Pyramidal and chiral groupings of gold nanocrystals assembled using DNA scaffolds,” J. Am. Chem. Soc. 131(24), 8455–8459 (2009). [CrossRef]  

53. A. Lakhtakia, Selected Papers on Natural Optical Activity (SPIE Optical Engineering, 1997).

54. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

55. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17(4), 2224–2234 (2009). [CrossRef]  

56. M. Nieto-Vesperinas, J. J. Saenz, R. Gomez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18(11), 11428–11443 (2010). [CrossRef]  

57. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011). [CrossRef]  

58. A. A. E. Saleh and J. A. Dionne, “Toward efficient optical trapping of sub-10-nm particles with coaxial plasmonic apertures,” Nano Lett. 12(11), 5581–5586 (2012). [CrossRef]  

59. A. Kotnala and R. Gordon, “Double nanohole optical tweezers visualize protein p53 suppressing unzipping of single DNA-hairpins,” Biomed. Opt. Express 5(6), 1886–1894 (2014). [CrossRef]  

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References

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  1. C. M. Dobson, “Protein folding and misfolding,” Nature 426(6968), 884–890 (2003).
    [Crossref]
  2. M. Quack, “How important is parity violation for molecular and biomolecular chirality?” Angew. Chem., Int. Ed. 41(24), 4618–4630 (2002).
    [Crossref]
  3. L. A. Nguyen, H. He, and C. Pham-Huy, “Chiral drugs: an overview,” Int. J. Biomed. Sci. 2(2), 85 (2006).
  4. G. Gübitz and M. G. Schmid, “Chiral separation by chromatographic and electromigration techniques,” Biopharm. Drug Dispos. 22(7-8), 291–336 (2001).
    [Crossref]
  5. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
    [Crossref]
  6. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref]
  7. M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4(1), 1768 (2013).
    [Crossref]
  8. A. Hayat, J. P. Balthasar Müller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
    [Crossref]
  9. A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
    [Crossref]
  10. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16(1), 013020 (2014).
    [Crossref]
  11. G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
    [Crossref]
  12. D. S. Bradshaw and D. L. Andrews, “Electromagnetic trapping of chiral molecules: orientational effects of the irradiating beam,” J. Opt. Soc. Am. B 32(5), B25–B31 (2015).
    [Crossref]
  13. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014).
    [Crossref]
  14. Y. Zhao, A. A. E. Saleh, and J. A. Dionne, “Enantioselective optical trapping of chiral nanoparticles with plasmonic tweezers,” ACS Photonics 3(3), 304–309 (2016).
    [Crossref]
  15. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
    [Crossref]
  16. N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12(2), 977–983 (2012).
    [Crossref]
  17. Z. G. Zheng, Y. Q. Lu, and Q. Li, “Photoprogrammable mesogenic soft helical architectures: a promising avenue toward future chiro-optics,” Adv. Mater. 32(41), 1905318 (2020).
    [Crossref]
  18. V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
    [Crossref]
  19. M. H. Alizadeh and B. M. Reinhard, “Plasmonically enhanced chiral optical fields and forces in achiral split ring resonators,” ACS Photonics 2(3), 361–368 (2015).
    [Crossref]
  20. T. Shoji and Y. Tsuboi, “Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostructure, light source, and resonant trapping,” J. Phys. Chem. Lett. 5(17), 2957–2967 (2014).
    [Crossref]
  21. X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
    [Crossref]
  22. K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2(1), 469 (2011).
    [Crossref]
  23. D. G. Kotsifaki and S. N. Chormaic, “Plasmonic optical tweezers based on nanostructures: fundamentals, advances and prospects,” Nanophotonics 8(7), 1227–1245 (2019).
    [Crossref]
  24. A. A. E. Saleh, S. Sheikhoelislami, S. Gastelum, and J. A. Dionne, “Grating-flanked plasmonic coaxial apertures for efficient fiber optical tweezers,” Opt. Express 24(18), 20593–20603 (2016).
    [Crossref]
  25. Z. H. Lin, J. Zhang, and J. S. Huang, “Plasmonic elliptical nanoholes for chiroptical analysis and enantioselective optical trapping”, arXiv 2101.04453 (2021).
  26. W. Zhang, T. Wu, R. Wang, and X. Zhang, “Surface-enhanced circular dichroism of oriented chiral molecules by plasmonic nanostructures,” J. Phys. Chem. C 121(1), 666–675 (2017).
    [Crossref]
  27. A. Pham, A. Zhao, C. Genet, and A. Drezet, “Optical chirality density and flux measured in the local density of states of spiral plasmonic structures,” Phys. Rev. A 98(1), 013837 (2018).
    [Crossref]
  28. L. D. Barron, Molecular Light Scattering and Optical Activity, second edition (Cambridge University, 2004).
  29. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010).
    [Crossref]
  30. Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011).
    [Crossref]
  31. H. K. Bisoyi and Q. Li, “Light-directed dynamic chirality inversion in functional self-organized helical superstructures,” Angew. Chem., Int. Ed. 55(9), 2994–3010 (2016).
    [Crossref]
  32. Y. Liu, W. Zhao, Y. Ji, R. Y. Wang, X. Wu, and X. D. Zhang, “Strong superchiral field in hot spots and its interaction with chiral molecules,” Europhys. Lett. 110(1), 17008 (2015).
    [Crossref]
  33. T. Wu, X. H. Zhang, R. Y. Wang, and X. D. Zhang, “Strongly enhanced Raman optical activity in molecules by magnetic response of nanoparticles,” J. Phys. Chem. C 120(27), 14795–14804 (2016).
    [Crossref]
  34. C. S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. A. Dionne, “Enhancing enantioselective absorption using dielectric nanospheres,” ACS Photonics 4(2), 197–203 (2017).
    [Crossref]
  35. G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
    [Crossref]
  36. W. Zhang, T. Wu, R. Wang, and X. Zhang, “Amplification of molecular chiroptical effect by low-loss dielectric nanoantennas,” Nanoscale 9(17), 5701–5707 (2017).
    [Crossref]
  37. T. Wu, W. Zhang, R. Wang, and X. Zhang, “A giant chiroptical effect caused by the electric quadrupole,” Nanoscale 9(16), 5110–5118 (2017).
    [Crossref]
  38. M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
    [Crossref]
  39. E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
    [Crossref]
  40. F. Graf, J. Feis, X. Garcia-Santiago, M. Wegener, C. Rockstuhl, and I. Fernandez-Corbaton, “Achiral, helicity preserving, and resonant structures for enhanced sensing of chiral molecules,” ACS Photonics 6(2), 482–491 (2019).
    [Crossref]
  41. J. Hu, M. Lawrence, and J. A. Dionne, “High quality factor dielectric metasurfaces for ultraviolet circular dichroism spectroscopy,” ACS Photonics 7(1), 36–42 (2020).
    [Crossref]
  42. P. Giovanni, F. Marco, C. Michele, D. Lamberto, A. I. Maria, M. M. Onofrio, and B. Paolo, “Superchiral surface waves for all-optical enantiomer separation,” J. Phys. Chem. C 123(46), 28336–28342 (2019).
    [Crossref]
  43. T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
    [Crossref]
  44. T. Wu, S. S. Hou, W. X. Zhang, and X. D. Zhang, “Strong superchiral fields and an ultrasensitive chiral sensor of biomolecules based on a dielectric photonic crystal slab with air holes,” Phys. Rev. A 102(5), 053519 (2020).
    [Crossref]
  45. A. Vazquez-Guardado and D. Chanda, “Superchiral light generation on degenerate achiral surfaces,” Phys. Rev. Lett. 120(13), 137601 (2018).
    [Crossref]
  46. I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111(6), 060401 (2013).
    [Crossref]
  47. S. Yu, H. S. Park, X. Piao, B. Min, and N. Park, “Low-dimensional optical chirality in complex potentials,” Optica 3(9), 1025–1032 (2016).
    [Crossref]
  48. S. S. Hou, T. Wu, W. X. Zhang, and X. D. Zhang, “Strongly enhanced Raman optical activity of chiral molecules by vector exceptional points,” J. Phys. Chem. C 124(45), 24970–24977 (2020).
    [Crossref]
  49. X. Wei, T. Tanaka, Y. Yomogida, N. Sato, R. Saito, and H. Kataura, “Experimental determination of excitonic band structures of single-walled carbon nanotubes using circular dichroism spectra,” Nat. Commun. 7(1), 12899 (2016).
    [Crossref]
  50. A. Ben-Moshe, S. G. Wolf, M. B. Sadan, L. Houben, Z. Fan, A. O. Govorov, and G. Markovich, “Enantioselective control of lattice and shape chirality in inorganic nanostructures using chiral biomolecules,” Nat. Commun. 5(1), 4302 (2014).
    [Crossref]
  51. K. Varga, S. Tannir, B. E. Haynie, B. M. Leonard, V. Dzyuba, J. Kubelka, and M. Balaz, “CdSe quantum dots functionalized with chiral, thiol-free carboxylic acids: unraveling structural requirements for ligand-induced chirality,” ACS Nano 11(10), 9846–9853 (2017).
    [Crossref]
  52. A. J. Mastroianni, S. A. Claridge, and A. P. Alivisatos, “Pyramidal and chiral groupings of gold nanocrystals assembled using DNA scaffolds,” J. Am. Chem. Soc. 131(24), 8455–8459 (2009).
    [Crossref]
  53. A. Lakhtakia, Selected Papers on Natural Optical Activity (SPIE Optical Engineering, 1997).
  54. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).
  55. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17(4), 2224–2234 (2009).
    [Crossref]
  56. M. Nieto-Vesperinas, J. J. Saenz, R. Gomez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18(11), 11428–11443 (2010).
    [Crossref]
  57. J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
    [Crossref]
  58. A. A. E. Saleh and J. A. Dionne, “Toward efficient optical trapping of sub-10-nm particles with coaxial plasmonic apertures,” Nano Lett. 12(11), 5581–5586 (2012).
    [Crossref]
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2020 (5)

Z. G. Zheng, Y. Q. Lu, and Q. Li, “Photoprogrammable mesogenic soft helical architectures: a promising avenue toward future chiro-optics,” Adv. Mater. 32(41), 1905318 (2020).
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T. Wu, W. X. Zhang, H. Z. Zhang, S. S. Hou, G. Y. Chen, R. B. Liu, C. C. Lu, J. F. Li, R. Y. Wang, P. F. Duan, J. J. Li, B. Wang, L. Shi, J. Zi, and X. D. Zhang, “Vector exceptional points with strong superchiral fields,” Phys. Rev. Lett. 124(8), 083901 (2020).
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T. Wu, S. S. Hou, W. X. Zhang, and X. D. Zhang, “Strong superchiral fields and an ultrasensitive chiral sensor of biomolecules based on a dielectric photonic crystal slab with air holes,” Phys. Rev. A 102(5), 053519 (2020).
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J. Hu, M. Lawrence, and J. A. Dionne, “High quality factor dielectric metasurfaces for ultraviolet circular dichroism spectroscopy,” ACS Photonics 7(1), 36–42 (2020).
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S. S. Hou, T. Wu, W. X. Zhang, and X. D. Zhang, “Strongly enhanced Raman optical activity of chiral molecules by vector exceptional points,” J. Phys. Chem. C 124(45), 24970–24977 (2020).
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2019 (4)

P. Giovanni, F. Marco, C. Michele, D. Lamberto, A. I. Maria, M. M. Onofrio, and B. Paolo, “Superchiral surface waves for all-optical enantiomer separation,” J. Phys. Chem. C 123(46), 28336–28342 (2019).
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F. Graf, J. Feis, X. Garcia-Santiago, M. Wegener, C. Rockstuhl, and I. Fernandez-Corbaton, “Achiral, helicity preserving, and resonant structures for enhanced sensing of chiral molecules,” ACS Photonics 6(2), 482–491 (2019).
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M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
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D. G. Kotsifaki and S. N. Chormaic, “Plasmonic optical tweezers based on nanostructures: fundamentals, advances and prospects,” Nanophotonics 8(7), 1227–1245 (2019).
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2018 (3)

A. Pham, A. Zhao, C. Genet, and A. Drezet, “Optical chirality density and flux measured in the local density of states of spiral plasmonic structures,” Phys. Rev. A 98(1), 013837 (2018).
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E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
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A. Vazquez-Guardado and D. Chanda, “Superchiral light generation on degenerate achiral surfaces,” Phys. Rev. Lett. 120(13), 137601 (2018).
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2017 (6)

K. Varga, S. Tannir, B. E. Haynie, B. M. Leonard, V. Dzyuba, J. Kubelka, and M. Balaz, “CdSe quantum dots functionalized with chiral, thiol-free carboxylic acids: unraveling structural requirements for ligand-induced chirality,” ACS Nano 11(10), 9846–9853 (2017).
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C. S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. A. Dionne, “Enhancing enantioselective absorption using dielectric nanospheres,” ACS Photonics 4(2), 197–203 (2017).
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G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
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W. Zhang, T. Wu, R. Wang, and X. Zhang, “Amplification of molecular chiroptical effect by low-loss dielectric nanoantennas,” Nanoscale 9(17), 5701–5707 (2017).
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T. Wu, W. Zhang, R. Wang, and X. Zhang, “A giant chiroptical effect caused by the electric quadrupole,” Nanoscale 9(16), 5110–5118 (2017).
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W. Zhang, T. Wu, R. Wang, and X. Zhang, “Surface-enhanced circular dichroism of oriented chiral molecules by plasmonic nanostructures,” J. Phys. Chem. C 121(1), 666–675 (2017).
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2016 (6)

A. A. E. Saleh, S. Sheikhoelislami, S. Gastelum, and J. A. Dionne, “Grating-flanked plasmonic coaxial apertures for efficient fiber optical tweezers,” Opt. Express 24(18), 20593–20603 (2016).
[Crossref]

H. K. Bisoyi and Q. Li, “Light-directed dynamic chirality inversion in functional self-organized helical superstructures,” Angew. Chem., Int. Ed. 55(9), 2994–3010 (2016).
[Crossref]

Y. Zhao, A. A. E. Saleh, and J. A. Dionne, “Enantioselective optical trapping of chiral nanoparticles with plasmonic tweezers,” ACS Photonics 3(3), 304–309 (2016).
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X. Wei, T. Tanaka, Y. Yomogida, N. Sato, R. Saito, and H. Kataura, “Experimental determination of excitonic band structures of single-walled carbon nanotubes using circular dichroism spectra,” Nat. Commun. 7(1), 12899 (2016).
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S. Yu, H. S. Park, X. Piao, B. Min, and N. Park, “Low-dimensional optical chirality in complex potentials,” Optica 3(9), 1025–1032 (2016).
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T. Wu, X. H. Zhang, R. Y. Wang, and X. D. Zhang, “Strongly enhanced Raman optical activity in molecules by magnetic response of nanoparticles,” J. Phys. Chem. C 120(27), 14795–14804 (2016).
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2015 (5)

S. Wheaton and R. Gordon, “Molecular weight characterization of single globular proteins using optical nanotweezers,” Analyst 140(14), 4799–4803 (2015).
[Crossref]

D. S. Bradshaw and D. L. Andrews, “Electromagnetic trapping of chiral molecules: orientational effects of the irradiating beam,” J. Opt. Soc. Am. B 32(5), B25–B31 (2015).
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A. Hayat, J. P. Balthasar Müller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
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Y. Liu, W. Zhao, Y. Ji, R. Y. Wang, X. Wu, and X. D. Zhang, “Strong superchiral field in hot spots and its interaction with chiral molecules,” Europhys. Lett. 110(1), 17008 (2015).
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M. H. Alizadeh and B. M. Reinhard, “Plasmonically enhanced chiral optical fields and forces in achiral split ring resonators,” ACS Photonics 2(3), 361–368 (2015).
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2014 (7)

T. Shoji and Y. Tsuboi, “Plasmonic optical tweezers toward molecular manipulation: tailoring plasmonic nanostructure, light source, and resonant trapping,” J. Phys. Chem. Lett. 5(17), 2957–2967 (2014).
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S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014).
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R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16(1), 013020 (2014).
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G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
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J. Berthelot, S. S. Acimovic, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
[Crossref]

A. Kotnala and R. Gordon, “Double nanohole optical tweezers visualize protein p53 suppressing unzipping of single DNA-hairpins,” Biomed. Opt. Express 5(6), 1886–1894 (2014).
[Crossref]

A. Ben-Moshe, S. G. Wolf, M. B. Sadan, L. Houben, Z. Fan, A. O. Govorov, and G. Markovich, “Enantioselective control of lattice and shape chirality in inorganic nanostructures using chiral biomolecules,” Nat. Commun. 5(1), 4302 (2014).
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2013 (4)

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111(6), 060401 (2013).
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V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
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A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
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M. C. Zhong, X. B. Wei, J. H. Zhou, Z. Q. Wang, and Y. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. 4(1), 1768 (2013).
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2012 (2)

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12(2), 977–983 (2012).
[Crossref]

A. A. E. Saleh and J. A. Dionne, “Toward efficient optical trapping of sub-10-nm particles with coaxial plasmonic apertures,” Nano Lett. 12(11), 5581–5586 (2012).
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2011 (5)

J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
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Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011).
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X. Yang, Y. Liu, R. F. Oulton, X. Yin, and X. Zhang, “Optical forces in hybrid plasmonic waveguides,” Nano Lett. 11(2), 321–328 (2011).
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K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2(1), 469 (2011).
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M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011).
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2010 (2)

2009 (2)

A. J. Mastroianni, S. A. Claridge, and A. P. Alivisatos, “Pyramidal and chiral groupings of gold nanocrystals assembled using DNA scaffolds,” J. Am. Chem. Soc. 131(24), 8455–8459 (2009).
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P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17(4), 2224–2234 (2009).
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2006 (1)

L. A. Nguyen, H. He, and C. Pham-Huy, “Chiral drugs: an overview,” Int. J. Biomed. Sci. 2(2), 85 (2006).

2003 (2)

C. M. Dobson, “Protein folding and misfolding,” Nature 426(6968), 884–890 (2003).
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D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
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2002 (1)

M. Quack, “How important is parity violation for molecular and biomolecular chirality?” Angew. Chem., Int. Ed. 41(24), 4618–4630 (2002).
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2001 (1)

G. Gübitz and M. G. Schmid, “Chiral separation by chromatographic and electromigration techniques,” Biopharm. Drug Dispos. 22(7-8), 291–336 (2001).
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A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
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N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12(2), 977–983 (2012).
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Acimovic, S. S.

J. Berthelot, S. S. Acimovic, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
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Alivisatos, A. P.

A. J. Mastroianni, S. A. Claridge, and A. P. Alivisatos, “Pyramidal and chiral groupings of gold nanocrystals assembled using DNA scaffolds,” J. Am. Chem. Soc. 131(24), 8455–8459 (2009).
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Alizadeh, M. H.

M. H. Alizadeh and B. M. Reinhard, “Plasmonically enhanced chiral optical fields and forces in achiral split ring resonators,” ACS Photonics 2(3), 361–368 (2015).
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Altug, H.

E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
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E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
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Balaz, M.

K. Varga, S. Tannir, B. E. Haynie, B. M. Leonard, V. Dzyuba, J. Kubelka, and M. Balaz, “CdSe quantum dots functionalized with chiral, thiol-free carboxylic acids: unraveling structural requirements for ligand-induced chirality,” ACS Nano 11(10), 9846–9853 (2017).
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Balthasar Müller, J. P.

A. Hayat, J. P. Balthasar Müller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
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L. D. Barron, Molecular Light Scattering and Optical Activity, second edition (Cambridge University, 2004).

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V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013).
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A. Ben-Moshe, S. G. Wolf, M. B. Sadan, L. Houben, Z. Fan, A. O. Govorov, and G. Markovich, “Enantioselective control of lattice and shape chirality in inorganic nanostructures using chiral biomolecules,” Nat. Commun. 5(1), 4302 (2014).
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Berthelot, J.

J. Berthelot, S. S. Acimovic, M. L. Juan, M. P. Kreuzer, J. Renger, and R. Quidant, “Three-dimensional manipulation with scanning near-field optical nanotweezers,” Nat. Nanotechnol. 9(4), 295–299 (2014).
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Biagioni, P.

G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
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Bisoyi, H. K.

H. K. Bisoyi and Q. Li, “Light-directed dynamic chirality inversion in functional self-organized helical superstructures,” Angew. Chem., Int. Ed. 55(9), 2994–3010 (2016).
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Bradshaw, D. S.

Brasselet, E.

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5(1), 3577 (2014).
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Cameron, R. P.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16(1), 013020 (2014).
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Canaguier-Durand, A.

A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
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Capasso, F.

A. Hayat, J. P. Balthasar Müller, and F. Capasso, “Lateral chirality-sorting optical forces,” Proc. Natl. Acad. Sci. U. S. A. 112(43), 13190–13194 (2015).
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G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
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Chan, C. T.

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5(1), 3307 (2014).
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J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
[Crossref]

Chanda, D.

A. Vazquez-Guardado and D. Chanda, “Superchiral light generation on degenerate achiral surfaces,” Phys. Rev. Lett. 120(13), 137601 (2018).
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Chaumet, P. C.

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J. Chen, J. Ng, Z. F. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics 5(9), 531–534 (2011).
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D. G. Kotsifaki and S. N. Chormaic, “Plasmonic optical tweezers based on nanostructures: fundamentals, advances and prospects,” Nanophotonics 8(7), 1227–1245 (2019).
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Claridge, S. A.

A. J. Mastroianni, S. A. Claridge, and A. P. Alivisatos, “Pyramidal and chiral groupings of gold nanocrystals assembled using DNA scaffolds,” J. Am. Chem. Soc. 131(24), 8455–8459 (2009).
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Cohen, A. E.

Y. Tang and A. E. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science 332(6027), 333–336 (2011).
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Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104(16), 163901 (2010).
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K. Wang, E. Schonbrun, P. Steinvurzel, and K. B. Crozier, “Trapping and rotating nanoparticles using a plasmonic nano-tweezer with an integrated heat sink,” Nat. Commun. 2(1), 469 (2011).
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E. Mohammadi, K. L. Tsakmakidis, A. N. Askarpour, P. Dehkhoda, A. Tavakoli, and H. Altug, “Nanophotonic platforms for enhanced chiral sensing,” ACS Photonics 5(7), 2669–2675 (2018).
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J. Hu, M. Lawrence, and J. A. Dionne, “High quality factor dielectric metasurfaces for ultraviolet circular dichroism spectroscopy,” ACS Photonics 7(1), 36–42 (2020).
[Crossref]

M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
[Crossref]

C. S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. A. Dionne, “Enhancing enantioselective absorption using dielectric nanospheres,” ACS Photonics 4(2), 197–203 (2017).
[Crossref]

A. A. E. Saleh, S. Sheikhoelislami, S. Gastelum, and J. A. Dionne, “Grating-flanked plasmonic coaxial apertures for efficient fiber optical tweezers,” Opt. Express 24(18), 20593–20603 (2016).
[Crossref]

Y. Zhao, A. A. E. Saleh, and J. A. Dionne, “Enantioselective optical trapping of chiral nanoparticles with plasmonic tweezers,” ACS Photonics 3(3), 304–309 (2016).
[Crossref]

A. A. E. Saleh and J. A. Dionne, “Toward efficient optical trapping of sub-10-nm particles with coaxial plasmonic apertures,” Nano Lett. 12(11), 5581–5586 (2012).
[Crossref]

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C. M. Dobson, “Protein folding and misfolding,” Nature 426(6968), 884–890 (2003).
[Crossref]

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A. Pham, A. Zhao, C. Genet, and A. Drezet, “Optical chirality density and flux measured in the local density of states of spiral plasmonic structures,” Phys. Rev. A 98(1), 013837 (2018).
[Crossref]

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[Crossref]

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G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
[Crossref]

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A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[Crossref]

Dzyuba, V.

K. Varga, S. Tannir, B. E. Haynie, B. M. Leonard, V. Dzyuba, J. Kubelka, and M. Balaz, “CdSe quantum dots functionalized with chiral, thiol-free carboxylic acids: unraveling structural requirements for ligand-induced chirality,” ACS Nano 11(10), 9846–9853 (2017).
[Crossref]

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A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15(12), 123037 (2013).
[Crossref]

Fan, Z.

A. Ben-Moshe, S. G. Wolf, M. B. Sadan, L. Houben, Z. Fan, A. O. Govorov, and G. Markovich, “Enantioselective control of lattice and shape chirality in inorganic nanostructures using chiral biomolecules,” Nat. Commun. 5(1), 4302 (2014).
[Crossref]

N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12(2), 977–983 (2012).
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F. Graf, J. Feis, X. Garcia-Santiago, M. Wegener, C. Rockstuhl, and I. Fernandez-Corbaton, “Achiral, helicity preserving, and resonant structures for enhanced sensing of chiral molecules,” ACS Photonics 6(2), 482–491 (2019).
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F. Graf, J. Feis, X. Garcia-Santiago, M. Wegener, C. Rockstuhl, and I. Fernandez-Corbaton, “Achiral, helicity preserving, and resonant structures for enhanced sensing of chiral molecules,” ACS Photonics 6(2), 482–491 (2019).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, N. Tischler, X. Vidal, M. L. Juan, and G. Molina-Terriza, “Electromagnetic duality symmetry and helicity conservation for the macroscopic maxwell’s equations,” Phys. Rev. Lett. 111(6), 060401 (2013).
[Crossref]

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G. Pellegrini, M. Finazzi, M. Celebrano, L. Duò, and P. Biagioni, “Chiral surface waves for enhanced circular dichroism,” Phys. Rev. B 95(24), 241402 (2017).
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N. A. Abdulrahman, Z. Fan, T. Tonooka, S. M. Kelly, N. Gadegaard, E. Hendry, A. O. Govorov, and M. Kadodwala, “Induced chirality through electromagnetic coupling between chiral molecular layers and plasmonic nanostructures,” Nano Lett. 12(2), 977–983 (2012).
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C. S. Ho, A. Garcia-Etxarri, Y. Zhao, and J. A. Dionne, “Enhancing enantioselective absorption using dielectric nanospheres,” ACS Photonics 4(2), 197–203 (2017).
[Crossref]

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M. L. Solomon, J. Hu, M. Lawrence, A. García-Etxarri, and J. A. Dionne, “Enantiospecific optical enhancement of chiral sensing and separation with dielectric metasurfaces,” ACS Photonics 6(1), 43–49 (2019).
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F. Graf, J. Feis, X. Garcia-Santiago, M. Wegener, C. Rockstuhl, and I. Fernandez-Corbaton, “Achiral, helicity preserving, and resonant structures for enhanced sensing of chiral molecules,” ACS Photonics 6(2), 482–491 (2019).
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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. (a) Schematic diagram of the PhC slab placed on the SiO2 substrate. There is a gap between the PhC slab and the substrate. (b) Geometry and coordinate of a unit cell for the PhC slab. The lattice constant, the radius of the nanohole and the thickness of the PhC slab are marked by a, r and t, respectively. (c) Dispersion relation in ΓΧ and ΓΜ directions as a function of |k|a/2π. Where k is the Bloch wave vector. The structural parameters are chosen as a = 395 nm, r = 60 nm, t = 280 nm. Moreover, c is the vacuum speed of light. (d) Dispersion relation in ΓΧ and ΓΜ directions as a function of Bloch wave vector normalized by 2π/a. The structural parameters are chosen as a = 395 nm, r = 60 nm, t = 326.2 nm. (e)-(f) The distributions of electric ($\widetilde {\textbf E}$) and magnetic ($\widetilde {\textbf H}$) fields for the double degenerate TM-like and TE-like modes in the YZ plane of the unit cell, respectively. The blue arrows denote the real part of the electric and magnetic fields. In addition, the black solid line frames the Si3N4 of the unit cell. The parameters of the PhC slab are chosen as a = 395 nm, r = 60 nm, and t = 280 nm. (g)-(i) The averaged enhancements of electric, magnetic and optical chirality in the cylindrical nanohole of the PhC slab with the cases of t = 280 nm and 326.2 nm.
Fig. 2.
Fig. 2. (a) Numerically calculated the total optical force F and the gradient force Fg acting on the chiral particles with the excitation of LCP light. The radius of the chiral particle is 10 nm. Moreover, the chiral particle is placed 20 nm above the PhC slab. (b) The magnitude of dielectric gradient force Fd and chiral gradient force Fk with the thickness of the PhC slab t = 326.2 nm. (c) The magnitude of dielectric gradient force Fd and chiral gradient force Fk with the thickness of the PhC slab t = 280 nm. The radius of the chiral particle is 10 nm with the refractive index nr = 1.45 and the particle chirality is κ = +0.5. The other structural parameters are chosen as a = 336 nm, r = 80 nm.
Fig. 3.
Fig. 3. (a) and (b) Calculated optical forces for both enantiomers (κ = ±0.5) in the unit cell with the excitation of LCP light. The gray areas represent Si3N4 material. Moreover, the black solid line frames the Si3N4 of the unit cell. The non-dielectric regions include the nanohole and the range of 360 nm above and below the unit cell. The color map shows the magnitude of the optical force, and the white arrows show the directions of the optical force. The optical force along the z-axis (-z-axis) is marked by the upward (downward) arrows. The excitation wavelengths with the thickness of the PhC slab (a) t = 326.2 nm and (b) t = 280 nm are 719.8 nm and 705.2 nm, respectively. (c) The trapping potential along the z-axis for both enantiomers when the thickness of the PhC slab is 326.2 nm under the excitation of LCP light. Here, the excitation wavelength is 719.8 nm. (d) The trapping potential along the z-axis for both enantiomers when the thickness of the PhC slab is 280 nm under the excitation of LCP light. Here, the excitation wavelength is 705.2 nm. The red solid and blue dashed lines represent the trapping potential of the chiral particle with the chirality being 0.5 and −0.5, respectively. The trapping potential is calculated when the power of incident light is 100 mW in the unit cell.
Fig. 4.
Fig. 4. (a) Near-field distribution of the gradient of optical chirality with the thickness of the PhC slab t = 326.2 nm at the excitation wavelength λ = 719.8 nm. (b) Near-field distribution of the gradient of optical chirality with the thickness of the PhC slab t = 280 nm at the excitation wavelength λ = 705.2 nm. The color map shows the magnitude of the gradient of optical chirality $\nabla C$ and the red arrows denote the directions of $\nabla C$.
Fig. 5.
Fig. 5. (a) Optical force F and its components Fd and Fk acting on the chiral particle as a function of chirality κ under the excitation of LCP light at λ = 719.8 nm. The thickness of the PhC slab is 326.2 nm. (b) Optical force F and its components Fd and Fk acting on the chiral particle as a function of chirality κ under the excitation of LCP light at λ = 705.2 nm. The thickness of the PhC slab is 280 nm. The red solid, orange long dash and blue dash lines correspond to the forces F, Fd and Fk respectively. Other parameters in Fig. 5 are the same as Fig. 2. Here, the sign of the force represents its direction. The positive (negative) sign means that the direction of the force is along the + z-axis (–z-axis).

Equations (11)

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$$\left[ {\begin{array}{c} \textbf{p}\\ \textbf{m} \end{array}} \right] = \left[ {\begin{array}{cc} {{\alpha_{\textrm{ee}}}}&{i{\alpha_{\textrm{em}}}}\\ { - i{\alpha_{\textrm{em}}}}&{{\alpha_{\textrm{mm}}}} \end{array}} \right]\left[ {\begin{array}{c} \textbf{E}\\ \textbf{H} \end{array}} \right],$$
$${\alpha _{\textrm{em}}} ={-} 12\pi r_\textrm{p}^3\frac{{\kappa \sqrt {{\mu _0}{\varepsilon _0}} }}{{({\varepsilon _\textrm{r}} + 2{\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}},$$
$${\alpha _{\textrm{ee}}} = 4{\varepsilon _0}\pi r_\textrm{p}^3\frac{{({\varepsilon _\textrm{r}} - {\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}}{{({\varepsilon _\textrm{r}} + 2{\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}}, $$
$${\alpha _{\textrm{mm}}} ={-} 4{\mu _0}\pi r_\textrm{p}^3\frac{{{\kappa ^2}}}{{({\varepsilon _\textrm{r}} + 2{\varepsilon _{\textrm{rm}}})({\mu _\textrm{r}} + 2) - {\kappa ^2}}}. $$
$$\left\langle \textbf{F} \right\rangle = \frac{1}{2}{\textrm{Re}} \left[ {(\nabla {\textbf{E}^ \ast }) \cdot \textbf{p + (}\nabla {\textbf{H}^ \ast }) \cdot \textbf{m - }\frac{{c{k^4}}}{{6\pi \sqrt {{\varepsilon_{\textrm{rm}}}} }}(\textbf{p} \times {\textbf{m}^ \ast })} \right] .$$
$$\begin{array}{l} \left\langle \textbf{F} \right\rangle = \nabla U + \sigma \frac{{\left\langle \textbf{S} \right\rangle }}{c} - {\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{em}}}]\nabla \times \left\langle \textbf{S} \right\rangle + \frac{c}{{{\varepsilon _{\textrm{rm}}}}}{\sigma _\textrm{e}}\nabla \times \left\langle {{\textbf{L}_\textrm{e}}} \right\rangle + \frac{c}{{{\varepsilon _{\textrm{rm}}}}}{\sigma _\textrm{m}}\nabla \times \left\langle {{\textbf{L}_\textrm{m}}} \right\rangle \\ + \omega {\gamma _\textrm{e}}\left\langle {{\textbf{L}_\textrm{e}}} \right\rangle + \omega {\gamma _\textrm{m}}\left\langle {{\textbf{L}_\textrm{m}}} \right\rangle \textrm{ + }\frac{{c{k^4}}}{{12\pi \sqrt {{\varepsilon _{\textrm{rm}}}} }}{\kern 1pt} {\mathop{\rm Im}\nolimits} [{\alpha _{\textrm{ee}}}\alpha _{\textrm{mm}}^ \ast ]{\mathop{\rm Im}\nolimits} [\textbf{E} \times {\textbf{H}^ \ast }] \end{array},$$
$${\textbf{F}_\textrm{g}} = \frac{1}{4}({\textrm{Re}} [{\alpha _{\textrm{ee}}}]\nabla {|\textbf{E} |^2} + {\textrm{Re}} [{\alpha _{\textrm{mm}}}]\nabla {|\textbf{H} |^2} - 2{\textrm{Re}} [{\alpha _{\textrm{em}}}]\nabla {\mathop{\rm Im}\nolimits} [\textbf{H} \cdot {\textbf{E}^ \ast }]),$$
$${\textbf{F}_\textrm{d}} = \frac{1}{4}{\textrm{Re}} [{\alpha _{\textrm{ee}}}]\nabla {|\textbf{E} |^2} + \frac{1}{4}{\textrm{Re}} [{\alpha _{\textrm{mm}}}]\nabla {|\textbf{H} |^2} , $$
$${\textbf{F}_\textrm{k}} ={-} \frac{{{\textrm{Re}} [{\alpha _{\textrm{em}}}]}}{2}\nabla {\mathop{\rm Im}\nolimits} [\textbf{H} \cdot {\textbf{E}^ \ast }]. $$
$$\nabla C ={-} \frac{{{\varepsilon _0}{\mu _0}\omega }}{2}\nabla {\mathop{\rm Im}\nolimits} (\textbf{E}_x^ \ast{\cdot} {\textbf{H}_x} + \textbf{E}_y^ \ast{\cdot} {\textbf{H}_y} + \textbf{E}_z^ \ast{\cdot} {\textbf{H}_z}). $$
$$\left\langle \textbf{F} \right\rangle \approx {\textbf{F}_\textrm{g}} = {\textbf{F}_\textrm{d}} + {\textbf{F}_\textrm{k}}$$